Abstract
In this paper, we apply the nearly exact discretization schemes to propose a discrete model for the FitzHugh–Nagumo model. We show that the discrete model obtained preserves the dynamics and the known features of the continuous FitzHugh–Nagumo model. We do so by performing distance-based and probability-based similarity analysis. Additionally, a sensitivity analysis is also performed to analyze the most influential parameters of the discrete system.
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Bertram, R., Manish, T., Butte, J., Kiemel, T., Sherman, A.: Topological and phenomenological classification of bursting oscillations. Bull. Math. Biol. 57, 413–439 (1995)
Corinto, F., Lanza, V., Ascoli, A., Gilli, M.: Synchronization in networks of fitzhugh–nagumo neurons with memristor synapses. In: 20th European Conference on Circuit Theory and Design (ECCTD). IEEE, pp. 608–611 (2011)
De Angelis, F., De Angelis, M.: On solutions to a fitzhugh-rinzel type model. Ricerche di Matematica (2020). https://doi.org/10.1007/s11587-020-00483-y
De Angelis, M.: A priori estimates for excitable models. Meccanica 48, 2491–2496 (2013)
De Angelis, M., Renno, P.: Asymptotic effects of boundary perturbations in excitable systems. Discrete Contin. Dyn. Syst. Ser. B 19, 2039–2045 (2014)
Elaydi, S.N.: Discrete Chaos. Chapman & Hall/CRC, London (2007)
Elmer, C.E., Van Vleck, E.S.: Spatially discete fitzhugh-nagumo equations. SIAM J. Appl. Math. 65, 1153–1174 (2005)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445–466 (1961)
Foroutan, M., Manafian, J.: Taghipour-Farshi: Exact solutions for fitzhugh-nagumo model of nerve excitation via kudryashov method. Opt. Quant. Electron. (2017). https://doi.org/10.1007/s11082-017-1197-y
Foweraker, J.P.A., Brown, D., Marrs, R.W.: Discrete-time stimulation of the oscillatory and excitable forms of a fitzhugh-nagumo model applied to the pulsatile release of luteinizing hormone releasing hormone. Chaos: Interdiscip. J. Nonlinear Sci. 5, 200–208 (1995)
Grassetti, F., Gusowska, M., Michetti, E.: A dynamically consistent discretization method for goodwin model. Chaos Solitons Fractals 130, 109420 (2020)
Hodgkin, A.L., Huxley, A.F.: A quantitative description of membrane currents and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544 (1952)
Hollander, M., Wolfe, D. A.: Nonparametric statistical methods. Wiley Series in Probability and Mathematical Statistics, Wiley, New York-London-Sydney (1999)
Hupkes, H.J., Sandstede, B.: Traveling pulses for the discrete fitzhugh-nagumo system. SIAM J. Appl. Math. 9, 827–882 (2010)
Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. The MIT Press, Cambridge (2007)
Jing, Z., Chang, Y., Guo, B.: Bifurcation and chaos in discrete fitzhugh–nagumo system. Chaos solitons fractals 21, 701–720 (2004)
Juzekaeva, E., Nasretdinov, A., Battistoni, S., Berzina, S., Iannotta, S., Khazipov, R., Erokhin, V., Mukhtarov, M.: Coupling cortical neurons through electronic memristive synapse. Adv. Mater. Technol. 4, 1800350 (2019)
Kudryashov, N.A.: Asymptotic and exact solutions of the fitzhugh-nagumo model. Regul. Chaotic Dyn. 23, 152–160 (2018)
Kudryashov, N.A., Rybka, R.B., Sboev, A.G.: Analytical properties of the perturbed fitzhugh-nagumo model. Appl. Math. Lett. 76, 142–147 (2018)
Kwessi, E., Elaydi, S., Dennis, B., Livadiotis, G.: Nearly exact discretization of single species population models. Nat. Resour. Model. 31, e12167 (2018)
Mickens, R.E.: Advances in the applications of nonstandard finite difference scheme, Hackensack. World Scientific, NJ (2005)
Mickens, R.E.: Difference equations: Theory, applications, and advanced topics, Monographs and Research Notes in Mathematics, xxii+553, third ed. CRC Press, Boca Raton (2015)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962)
Quateroni, A., Manzoni, A., Vergara, C.: The cardiovascular system: mathematical modelling, numerical algorithms and clinical applications. Acta Numer. 26, 365–590 (2017)
Saltelli, A., Tarantola, S., Chan, K.-S.: A quantitative model-independent method for global sensitivity analysis of model output. Technometrics 41, 39–56 (1999)
Sehgal, S., Foulkes, A.J.: Numerical analysis of subcritical hopf bifurcations in the two-dimensional fitzhugh-nagumo model. Phys. Rev. E 102, 012212 (2020)
Sobol, I.M.: Sensitivity estimates for nonlinear mathematical models. Mat. Modelirovanie 2, 112–118 (1990)
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Kwessi, E., Edwards, L.J. A Nearly Exact Discretization Scheme for the FitzHugh–Nagumo Model. Differ Equ Dyn Syst 32, 253–275 (2024). https://doi.org/10.1007/s12591-021-00569-5
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DOI: https://doi.org/10.1007/s12591-021-00569-5